A Preemptive Goal Programming Model for the Sustainability of Growth in Engineering Colleges Elif Kongar* Departments of Mechanical Engineering and Technology Management. University of Bridgeport Tarek Sobh University of Bridgeport
Abstract Today, ever-decreasing budgets and dynamic variations in the number of both faculty and student bodies are two major challenges that most U.S. universities deal with. In addition to the effort to solve these problems, every higher education institution also concentrates on ensuring its sustainability in the academic and industrial arenas while providing continuous improvements in academic programming, resources and student and Faculty support services. To ensure sustainability and continuous growth, there are various additional parameters that need to be taken into account, such as: revenue from various sources, including tuition, grants from industry and government, and alumni/other donations. Enhancing the technological ability of the University via additional equipment purchases and/or maximizing the utilization of existing technologies can also be counted among the goals of any university. Mathematically, this administrative problem can be addressed using multiple objective modeling techniques. Goal Programming (GP) is a linear programming-based technique that has the ability to handle conflicting objectives in both preemptive and weighted manners. In this paper, we present a preemptive goal programming model for the School of Engineering at the University of Bridgeport. Data and case studies are provided along with a list of objectives for the Engineering School. Keywords: School of Engineering, Enrollment, Sustainability, Quality of Education, Preemptive Goal Programming, Multiple Criteria Optimization. 1. Introduction Today, ever-decreasing budgets and dynamic variations in the number of both faculty and student bodies are two major challenges that most U.S. universities deal with. In addition to the effort to solve these problems, every higher education institution also concentrates on ensuring its sustainability in the academic and industrial arenas while providing continuous improvements in academic programming, resources and student and Faculty support services. In order to ensure sustainability and continuous growth, there are various additional parameters that need to be taken into account, such as: revenue from various sources, including tuition, grants from industry and government, and alumni/other donations. Enhancing the technological ability of the University via additional equipment purchases and/or maximizing the utilization of existing technologies can also be counted among the goals of any university. Mathematically, this administrative problem can be addressed using multiple objective modeling techniques.
*
Corresponding Author. University of Bridgeport, 221 University Avenue, School of Engineering, 41 Technology Building, Bridgeport, CT 06604. Office: (203) 576-4379, Fax: (203) 576-4750, e-mail :
[email protected].
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Goal Programming (GP) is a linear programming-based technique that has the ability to handle conflicting objectives in both preemptive and weighted manners. In this paper, we present a preemptive goal programming model for the School of Engineering at the University of Bridgeport. Data and case studies are provided along with a list of objectives for the Engineering School. The paper is organized as follows: Introduction to the proposed Preemptive Goal Programming methodology is provided in the following section. Model development is the focus of Section 3. Case study data and modeling is provided in Section 4. The paper concludes with considerations regarding future enhancements. 2. Introduction to the Preemptive Goal Programming Model A large number of real world decision-making and optimization problems are actually multi-objective. Even so, many important optimization models, such as linear programming models, require that the decision maker express his/her wishes as one aggregate objective function that is usually subjected to some constraints. Goal programming (GP), generally applied to linear problems, deals with the achievement of prescribed goals or targets. First reported by Charnes and Cooper1, 2, it was then extended in the 1960s and 1970s by Ijiri3, Lee4 and Ignizio5. Since then, there has been an explosion of areas where goal programming has been applied. Both academicians and practitioners have embraced this technique. The basic purpose of goal programming is to simultaneously satisfy several goals relevant to the decision-making situation. To this end, a set of attributes to be considered in the problem situation is established. Then, for each attribute, a target value (i.e., appraisal level) is determined. Next, the deviation variables are introduced. These deviation variables may be negative or positive (represented by ηi and ρi, respectively). The negative deviation variable, ηi, represents the quantification of the under-achievement of the ith goal. Similarly, ρi represents the quantification of the over-achievement of the ith goal6. Finally for each attribute, the desire to overachieve (minimize ηi) or underachieve (minimize ρi), or satisfy the target value exactly (minimize ηi + ρi) is articulated6. Steps for the Preemptive Goal Programming algorithm is provided in Table 1. Figure 1 depicts the flow chart of the overall algorithm. Table 1. Preemptive Goal Programming Algorithm Step 1 2 3
4
5
Action Embed the relevant data set. Set the first goal set as the current goal set. Obtain a Linear Programming (LP) solution defining the current goal set as the objective function. If the current goal set is the final goal set, a. set it equal to the LP objective function value obtained in Step 2, and STOP. Otherwise, go to Step 4. If the current goal set is achieved or overachieved a. set it equal to its aspiration level and add the constraint to the constraint set, Go to Step 5. b. Otherwise, if the value of the current goal set is underachieved, set the aspiration level of the current goal equal to the LP objective function value obtained in Step 2. Add this equation to the constraint set. Go to Step 5. Set the next goal set of importance as the current goal set. Go to Step 2.
2
START
Step 1 Embed the relevant data set. Set the first goal set as the current goal set.
Step 2 Obtain a Linear Programming (LP) solution defining the current goal set as the objective function.
Yes
Step 3 Is current goal set the final goal set?
Step 4.a. Set the goal set equal to its aspiration level and add the constraint to the constraint set.
Yes
Step 4 Is current goal set achieved or overachieved?
No
No
Step 3.a. Obtain a Linear Programming (LP) solution defining the current goal set as the objective function.
Step 4.b. Set the aspiration level of the current goal equal to the LP objective function value obtained in Step 2. Add this equation to the constraint set.
Step 5 Set the next goal set of importance as the current goal set.
STOP
Figure 1. Flowchart of the Preemptive Goal Programming Algorithm 3. Model for the Sustainability of Growth in Engineering Colleges In this section we introduce the proposed model, including its objective function, technological constraints, and sign restrictions. 3.1. Goals of the Preemptive Goal Programming Model Goals for the preemptive GP model are provided in Table 2 along with their target, current, and tolerable limits. The priority of each goal is also provided in the table. The proposed algorithm aims at finding the number of full time faculty members (xf), the number of part time faculty members (xp), and the number of new students (xn).
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Table 2. Goals for the Preemptive Goal Programming Model Objective
Goal
Max Max Max Max Max Max Max Max Max Max Max Max Max Max Max Max Max Min Max Max RO* Max Max Max Max Max Max Max Max Max Min Max
# of Journal Publications/year Revenue from Research Grants Revenue from Non-Research Related Activities Student Enrollment # of Faculty Members (Full time faculty) Revenue from Tuition and Fees Faculty salaries (Current average, all) Quality of new students (Average incoming GPA) Students Graduation GPA (Average) Student employment percentage (job within a year) Conference/Workshop Chairmanships Journal/Book/Editorial Duties Technical Committee memberships Student Competition Participants Women Faculty # of Conference Publications/year Women Students Attrition Rate (Max Retention) Co-op and internship participation in co-op programs # of Staff (Administrative Personnel) # of Students per Class (Average) # of Projects sponsored by industry/year Faculty professional development funding #GA’s and RA’s offered per semester (credit hours) Student professional development funding Staff salaries (current average, all without Dean) Session Chairmanships Tech-related expenditure (s/w, h/w, etc.) # online courses offered/year # of New courses/semester Equivalent # of part-time faculty Actual # of part-time faculty
Target (Desirable) 52.5 $2.1M $6M 1500 36 $16M $90K 3.1 3.6 100% 11 21 42 130 12 84 260 3% 95% 10 25 50 $125K 1170 $65K $65K 42 $4M 50 15 10 60
*RO = Range Optimization
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Current
Tolerable
Priority
31 $0.8M $3.3M 1350 21 $14.8M $83K 2.87 3.35 97% 9 12 32 61 4 76 197 5% 86% 5 35 30 $87K 360 $26K $61K 24 $2.7M 28 * 17 21 48
21 * $1.05M * $4.0M 1200 19 $13M * $85K * 3.0 3.1 95% 7 10.5 21 * 65 *8 63 182 (14%) 7.5% 80% 5 * 30 25 $75K 300 $25K $61K 21 $2.0M 25 10 25 38
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
The model is applied to a multi-period environment using mixed integer goal programming (GP) method, which allows the introduction of the various priority levels. This GP can be described as follows: Find [xf, xp, xn, xs] so as to:
⎡ ⎤ ⎢ lexmin u = ( ∑ ni + ∑ gi ), (ηi , ρi )⎥ . ⎢ ∀hard ⎥ ∀hard ⎢⎣ const . ⎥⎦ const . where ( ∑ ni + ∑ g i ) represents the hard (technological) constraints of the model, and (η k , ρ k ) ∀hard const .
∀hard const .
represents the goals of the model and, xf = number of new Full Time faculty, xp = number of new Part Time faculty, xs = number of new staff, xt = number of new students. The first goal is to maximize the number of journal publications per full time faculty member in a year. In goal programming terms, our desire for the total number of journals publications (jpub) would be to aspire for a total number of at least jpub* and exceed it as much as possible. Mathematically, this can be achieved by forcing the negative deviation (η1) from the predetermined value, jpub*, to secure a value equal to zero. As can easily be observed from Eq. 1, by placing no restrictions on the positive deviation variable (ρ1), the model would ignore the ceiling on variable jpub*. In other words, this goal guarantees a highest value for the number of publications while articulating a lower aspiration perimeter. min η1 s.t. jpub + η1 - ρ1 = jpub *.
(1)
The second goal is to maximize the Revenue from Research Grants (rgrt) . This goal can be expressed mathematically as: min η2 s.t. rgrt + η2 + ρ2 = rgrt *.
(2)
The third goal is to maximize the Revenue from non-Research Related Activities (nrgt) such as discounts, donations, fund raising activities: min η3 s.t. ngrt + η3 + ρ3 = ngrt *.
(3)
The fourth goal is to maximize Student Enrollment (tt). This goal set is given in Eq. 4. min η4 s.t. tt + η4 + ρ4 = tt *.
(4)
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The fifth goal is to maximize the number of Full Time faculty (tf). This goal set is provided in Eq. 5. min η5 s.t. tf + η5 + ρ5 = tf *.
(5)
This completes the introduction of goals in the model. Next, we provide detailed information regarding the technological constraints of the model. 3.2. Technological Constraints of the Preemptive Goal Programming Model The number of journal publications (jpub) is a function of the total number of Full Time faculty members (tf), and the number of journal publications per faculty member (jpf). Hence, related technological constraints can be introduced to the model as in as in Eq. 6. tf = xf + cf tf = tf*.
(15)
However, in order to preserve the preemptive structure of the proposed model, it is avoided at this point. The results of the fourth goal programming model is provided in Table 5. Table 5. Second Preemptive Goal Programming Model with Additional Technological Constraints Obj. fn.* η2 η3 η4 η5 jpub η1 0 0 0 0 0 54 min η2 *with additional technological constraints Eq. 13.
rgrt 2.42
8
ngrt 7.26
tt 1500
tf 36
prf 14.49
xf 15
xp 52
xs 15
xt 150
As can be observed from Table 4 after solving the four consecutive PGP models, all goals are achieved at their corresponding aspiration levels. This also implies that embedding Eq. 15 into the model is not required since the results would remain unchanged. 5. Conclusions and Future Research The proposed paper attempts to find “best” solutions to factors that would ensure sustainability of the School of Engineering at the University Bridgeport. In this regard, a Preemptive Goal Programming model is applied to the first five goals of the School. Even though it is mathematically cumbersome to formularize the relationships between the goals and model variables given that the model reflects reality, it provides interesting results depicting the effects of various goals on the remaining system variables and goals. Hence, the model can also be utilized as a cause-effect impact analysis tool to understand the sensitive relationships between the variables. In the future, all the goals of the School of Engineering can be embedded in the model and the model can be adjusted according to the changing variables.
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References [1] [2] [3] [4] [5] [6]
Charnes, A. and Cooper, W.W., Management Models and Industrial Applications of Linear Programming. 1961, New York: John Wiley and Sons. Charnes, A., Cooper, W.W., and Ferguson, R.O., Optimal Estimation of Executive Compensation by Linear Programming. Management Science, 1955. 1(2): p. 138-151. Ijiri, Y., Management Goals and Accounting for Control. 1965, Chicago, IL: Rand McNally. Lee, S.M., Goal Programming for Decision Analysis. 1972, Philadelphia, PA: Auerbach Publishers. Ignizio, J.P., Goal Programming and Extensions. Lexington Books. 1976, MA: D.C. Heath and Company. Romero, C., Handbook of Critical Issues in Goal Programming. Oxford. 1991: Pergamon Press.
Biographical Information Dr. Elif Kongar received her BS degree from the Industrial Engineering Department of Yildiz Technical University, Istanbul, Turkey, in June 1995. In June 1997, she received her MS degree in Industrial Engineering from the same university where, she was awarded full scholarship for graduate studies in the USA. She obtained her Ph.D. degree in June 2003. She has been a research associate in the Laboratory for Responsible Manufacturing (LRM) at Northeastern University since September 1999. She has also been employed as an Assistant Professor by Yildiz Technical University till February 2006. Dr. Kongar is currently an Assistant Professor at Bridgeport University. Her research interests include the areas of supply chain management, logistics, environmentally conscious manufacturing, product recovery, disassembly systems, production planning and scheduling and multiple criteria decision making. Dr. Tarek M. Sobh received the B.Sc. in Engineering degree with honors in Computer Science and Automatic Control from the Faculty of Engineering, Alexandria University, Egypt in 1988, and M.S. and Ph.D. degrees in Computer and Information Science from the School of Engineering, University of Pennsylvania in 1989 and 1991, respectively. He is currently the Vice President for Graduate Studies and Research and the Dean of the School of Engineering; a Professor of Computer Science, Computer Engineering, Mechanical Engineering and Electrical Engineering and the Founding Director of the interdisciplinary Robotics, Intelligent Sensing, and Control (RISC) Laboratory at the University of Bridgeport, Connecticut. Dr. Sobh has published over 170 refereed journal and conference papers, and book chapters; and chaired many international conferences and technical meetings within the areas of Robotics and Automation, Computer Vision, Discrete Event Systems, Active Sensing, Uncertainty Modeling, Engineering Education, Online Engineering, Electromechanical Prototyping, and Management of Engineering Projects.
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